L]ETTER~ AL NUOVO CIM~NTO

VOL. 23, N. 15

9 Dieembre 1978

Strong Gravitation and Elementary Particles. J~G

JY P~RNG

Department o/ Mechanical Engineering, National Taiwan University - Taipei, Taiwan (ricevuto 1'11 Maggio 1978; manoscritto revisionato ricevuto il 25 Settembre 1978)

The purpose of this paper is to calculate the strong gravitational constant b y a proposed model for electron and proton and show that strong gravitation plays a role in stability of elementary particles. Ross (1) has obtained values for electron and proton radii by postulating a principle according to which a particle assumes a radius such t h a t the effective gravitational force is zero at that radius, the predicted radii of electron and proton are 2.80- I0 -la cm and 0.79-10 -13 cm, respectively. Recently high-energy experiments showing t h a t the electron is pointlike (2) and proton is built b y m a n y pointlike objects thus disfavour Ross' principle. I n order to reconcile this principle with experiments, we propose a model for electron and proton. We note that fig. 1 and fig. 2 are indistinguishable with respect to the exterior Schwarzchild solution of Einstein's law of gravitation if the mass and charge denoted b y the shaded region in fig. 1 is equal to that of fig. 2. The radius predicted b y the Ross principle is shown in fig. 1.

Fig. 1.

(1) (|)

552

Fig. 2.

D. •. R o s s : Nuovo Cimento, 8 A , 603 (1972). G. CAVALLERI and G. SPI~ELLI: NUOVO Cimento, 39 B, 93 (1977).

STRONG

GRAVITATION

AND

55~

]~,LEMENTARY PARTICLES

I t is o b v i o u s t h a t if we t a k e fig. 2 as o u r m o d e l for e l e c t r o n a n d p r o t o n , a n d c o n s i d e r fig. 1 as a special case of fig. 2 t h e n Ross p r i n c i p l e c a n c o m p l y w i t h h i g h - e n e r g y e x p e r i m e n t s . I n t h e case of t h e e l e c t r o n t h e r e g i o n A is p o i n t l i k e , i n t h e case of p r o t o n t h e r e are m a n y p o i n t l i k e o b j e c t s d i s t r i b u t e d i n t h i s r e g i o n w i t h s p h e r i c a l s y m m e t r y , t h e m e a n r a d i u s of p r o t o n c h a r g e d i s t r i b u t i o n m e a s u r e d b y iV[CALLISTER a n d HOFSTADTER (a) is 0.78"10 -13 e m h e n c e i n t h e case of p r o t o n t h e r e g i o n A is n o t p o i n t l i k e . T h e r e g i o n B r e p r e s e n t s t h e r e g i o n of s t r o n g g r a v i t a t i o n a l field w h i c h h o l d s r e g i o n A t i g h t l y . T h i s p r o v i d e s a n e x p l a n a t i o n for t h e s t a b i l i t y of e l e c t r o n a n d p r o t o n . I n o r d e r t o e x p l a i n t h e cause of t h i s b i n d i n g we h a v e t h e b a s i c a s s u m p t i o n as follows: T h e s p a c e - t i m e s t r u c t u r e of t h e r e g i o n B (fig. 2) h a s t h e v e r y special p r o p e r t y t h a t t h e w h o l e r e g i o n h a s n o l e n g t h c o n t r a c t i o n a l o n g t h e d i r e c t i o n of r, i.e., z e r o - r a d i u s contraction. I n t h e f o l l o w i n g c a l c u l a t i o n t h e s t r u c t u r e of region A is n o t n e e d e d . T h e S e h w a r z c h i l d s o l u t i o n of E i n s t e i n ' s l a w of g r a v i t a t i o n c a n b e w r i t t e n i n t h e f o r m dsS = gooc~ dr2-- gn d r S - - r2 d122,

(1)

d O s = dO s -~ sin s 0 d ~ s .

F o r e l e c t r o n a n d p r o t o n (1) we h a v e , r e s p e c t i v e l y ,

goo= g ~ = (l_2Gm~

GeS~

CST

134Ts] d

go0

V(r) = r-l(gn-- 1 ) ,

= e x p [V(r)] ,

(2) 1 --

gll

ml~

G Mp ---, Cs

2ml q- ~m e x p [ - - 2mr] -q- ~ exp [-- 2mr] T r

Gg~

~=--,

C4

m--

m~ c ~

)_1 ,

--0.68-10 laem,

gS

--=

he

14.50.

A c c o r d i n g t o o u r m o d e l , G is N e w t o n ' s g r a v i t a t i o n a l c o n s t a n t o n l y if r > R 0, w h e r e R o is t h e r a d i u s of t h e p a r t i c l e defined b y o u r m o d i f i e d Ross' p r i n c i p l e , i n t h e case of r < R o, i.e. i n t h e r a n g e of s t r o n g g r a v i t y , we d e t e r m i n e G b y t h e c o n d i t i o n of stab i l i t y of t h e e l e c t r o n . Our b a s i c a s s u m p t i o n a n d m o d e l of e l e c t r o n i m p l y t h a t g~-~(eleetron)l . . . . = O ,

(3)

w h e r e r, = 2 . 8 0 . 1 0 -la e m (1), m o = 9.11-10 -28 g, e = 4 . 8 0 . 1 0 -1~ C.G.S. ,

c ~ 2.99-101~ c m / s .

W e get f r o m (3) (4)

(a)

G = 2 . 7 7 - 1 0 a5 C.G.S.

R. W. I~CALLISTER and R. HOFSTADTER: Phys. Rev., 102, 851 (1956).

554

J~O

J y PE~NO

In order to justify this new constant, we use the condition of stability of the proton to calculate the mass of the proton Mp. (5)

g~t~(proton) [~_rp= 0 ,

where rr = 0.79-10 -13 em (i). Substituting the d a t a of (2) and (4) in (5), we get .Mp = 1 . 6 7 . 1 0 -24 g .

Our theory implies t h a t Einstein's law of gravitation is also valid for strong gravitation if and only if the gravitational constant is given b y (4). Since electron and proton, along with their antiparticles, are t h e only stable massive elementary particles, hence our calculation is unique. W e state our modified Ross' principle as follows: A particle assumes a radius such t h a t the effective Newton's gravitational force is zero at t h a t radius which represents the range of strong gravitation binding the mass and charge with spherical s y m m e t r y in a small region at the center. Other theories of general r e l a t i v i t y and elementary particles based on t h e quark strong charge and dilatation invariant hypotheses have been developed by R~CAmI et al. (a), and CALDIROLA eg al. (a-7).

(') (a) (8) to (7)

E. RECAMI a n d P. CASTORI~'A: Lett. Nuovo Cimento, 15, 347 (1976). P. CALDIROLA, ~r PAVSIC a n d E. RECAMI: Phys. Left., 6 6 A , 9 (1978). p. CALDIROLA, M. PAVSIO a n d E. RECA.MI: R e p o r t I N F N ] A E - 7 7 ] l O ( F r a s e a t i , L ~ q . F . N . , 1977), a p p e a r i n Nuovo Cimento, B. P . CALDIaOLA a n d E. RECAMI: R e p o r t I N F N / A E - 7 8 / 7 ( F r a s c a t i , L N . F . N . , 1978)o